Average Error: 40.1 → 0.3
Time: 8.1s
Precision: 64
Internal Precision: 128
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.00016527788259669608:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{3} - 1}{\left(e^{x} \cdot e^{x} + \left(e^{x} + 1\right)\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{6} \cdot {x}^{2} + \frac{1}{2} \cdot x\right) + 1\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.1
Target39.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.00016527788259669608

    1. Initial program 0.1

      \[\frac{e^{x} - 1}{x}\]
    2. Using strategy rm

    3. Applied flip3--0.1

      \[\leadsto \frac{\color{blue}{\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)}}}{x}\]

    4. Applied associate-/l/0.1

      \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{x \cdot \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)\right)}}\]

    5. Simplified0.1

      \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{3} - 1}}{x \cdot \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)\right)}\]

    if -0.00016527788259669608 < x

    1. Initial program 60.0

      \[\frac{e^{x} - 1}{x}\]

    2. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)}\]
    3. Using strategy rm

    4. Applied associate-+r+0.4

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x + \frac{1}{6} \cdot {x}^{2}\right) + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00016527788259669608:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{3} - 1}{\left(e^{x} \cdot e^{x} + \left(e^{x} + 1\right)\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{6} \cdot {x}^{2} + \frac{1}{2} \cdot x\right) + 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019021 
(FPCore (x)
  :name "Kahan's exp quotient"

  :herbie-target
  (if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x))

  (/ (- (exp x) 1) x))